Stressed thin-film membrane islands

ABSTRACT

A structure including a support defining an opening, and a tensilely stressed thin-film membrane disposed to occlude the opening, the membrane contacting at least a portion of the support. The stressed membrane includes a material having a characteristic crack spacing greater than one-half of a minimum dimension of the membrane and less than ten times the minimum dimension. A structure including a support defining a opening having a minimum opening dimension, and a compressively stressed thin-film membrane disposed to occlude the opening, the membrane contacting at least a portion of the support. The stressed membrane includes a membrane material having a critical aspect ratio for buckling that is greater than a ratio of one-half of the minimum opening dimension to a thickness of the membrane, and the critical aspect ratio for buckling is less than a ratio of ten times the minimum opening dimension to the thickness of the membrane.

RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application60/505,547 filed Sep. 23, 2003, the entire disclosure of which is herebyincorporated by reference.

FIELD OF THE INVENTION

The invention relates generally to stressed membranes and specificallyto stressed thin-film membranes for which fluidic access is required toboth sides of the membrane.

BACKGROUND

Many commercial applications require the separation of two fluids, e.g.,gasses or liquids, by using a membrane. The membrane is selected tomediate the interaction of the two fluids. For example, in a hydrogenpurification system, the membrane may mediate the interaction of ahydrogen-rich stream at high pressure from a pure hydrogen stream atlower pressure. In this example, the membrane may be constructed from amaterial that allows hydrogen diffusion at a higher rate than othercomponents of the hydrogen-rich stream. In another example, a membranein a fuel cell may mediate the interaction of an oxygen-containing fluidwith a fuel-containing fluid. The fuel cell membrane may includemultiple layers that allow one or more types of ions to pass through themembrane to oxidize the fuel, while extracting electrical energy fromthat reaction.

To improve performance and decrease system size, it is often desirableto miniaturize membrane-containing systems. However, the materials thatcan readily be manufactured in a miniaturized fashion are not alwayscompatible with the materials that are optimal for membranefunctionality. Integration of these two materials sets may result insubstantial stresses in the membrane. Stress may be induced by, forexample, different thermal expansion rates of the membrane and asupporting structure.

A specific example of a need for miniaturization may be found in thearea of batteries and fuel cells. The proliferation of portableelectronics, including cellular telephones and laptop computers, hasincreased the demand on power storage devices, such as batteries. Fuelcells may be used to increase energy storage available in comparison tobatteries. The fuel cell system, however, must be miniaturized to fitwithin the small form-factors of existing batteries. One example of atype of fuel cell is a solid oxide fuel cell, which is known to havehigh efficiency. One common miniaturization technique employs silicon(Si) substrates and integrated circuit manufacturing technologies.Silicon expands at a rate of approximately 4 micrometers per meter perdegree Celsius (μm/m/° C.). Conventional solid oxide fuel cells usematerials that expand at a rate of approximately 10 μm/m/° C., andoperate at temperatures of about 800° C. Combining a conventional solidoxide fuel cell membrane with conventional silicon manufacturing maycause a significant expansion mismatch of approximately 0.5%, leading tohighly stressed membranes. Additional factors contributing to the stressmay include intrinsic stress of the thin film as deposited, tensile orcompressive stress induced by sintering or other thermal processing, andchemical modifications inducing tensile or compressive stresses. Highstresses in the thin-film membranes may cause mechanical failure of thefilm or the stress level may undesirably change material properties.

Design of fuel cell membranes, such as yttrium-stabilized zirconia (YSZ)on Si substrates, may require a free-standing YSZ thin film to stretchover a 1 millimeter (mm) to a 1 centimeter (cm) diameter. Thesemembranes may fail because these membranes may be pliable or prone tobuckling. Furthermore, YSZ membranes may also fail when cooled into thetensile state because of crack propagation.

SUMMARY

The invention relates to the formation of stressed thin-film membranessubstantially free of cracks, configured so that fluidic contact to bothsides of the membranes is possible.

In the particular case of a tensilely stressed film fully bonded to arigid support, channel fractures or cracks in a web-like pattern mayform in the film. A distance between adjacent cracks is observed not tobe random, but rather to cluster near a characteristic crack spacing.This phenomenon is well known in the field of thin film fracturemechanics.

Reducing lateral dimensions of the tensilely stressed film below thischaracteristic crack spacing helps prevent the film from cracking. Inorder to produce larger active areas of stressed material, arrays ofislands of stressed films may be formed.

The existing techniques of island formation cannot be readily adaptedfor the formation of tensilely stressed membranes, i.e., membranestructures that allow fluidic access to both sides of a film. Therequirement for fluidic contact means that stressed membranes cannot bedisposed over solid support structures.

In the particular case of a compressively stressed membrane, the lack ofa supporting structure may result in expansion and warping of themembrane. This warping may be undesirable for various applications, andmay also cause cracking of the membrane.

Providing good adhesion to the substrate such that the bonding energy islarger than the energy in the stressed material may help preventcompressive failure. However, this existing technique of improvedadhesion cannot be readily adapted for the formation of stressedmembranes.

A structure in accordance with the invention is a small tensilelystressed membrane that has only a slight overlap with a supportingstructure, i.e., a support grid. The total dimensions of the stressedmaterial, including freestanding and bonded areas, are designed to besufficiently small so that the membrane is unlikely to fracture intension. The allowable dimensions for a tensilely stressed membrane aredictated by the characteristic crack spacing. Materials at the bondinterface, i.e., the support grid, affect the characteristic crackspacing, and therefore also affect the allowed dimensions of themembrane.

In an embodiment, the available active area is increased by theconstruction of islands arranged such that the spacing between islandsis relatively small. This embodiment may be referred to as“pre-cracking” of the film because of the similarities between theintentional spacings and the cracks that would have formed if not forthe creation of the spacings. The support structure may be a supportgrid aligned under the spaces with a small overlap with the stressedfilm. The dimensions of the support structure are designed according todesign rules presented below so that the film is unlikely to crack intension.

Another structure in accordance with the invention is a smallcompressively stressed membrane that is bonded to the support material.The allowable dimensions of the free-standing area of the film aredesigned according to design rules presented below so that the membraneis unlikely to buckle in compression.

In one embodiment, a support grid is provided to which the membrane maybe bonded, and a method is provided for forming an appropriate sealaround an outer edge of the membrane. The support grid, including amaterial such as silicon-rich silicon nitride, helps provide stiffnessto the membrane structure, as well as reduces the probability ofwarping. A small island of stressed thin-film material is rigidly bondedto a support material over an annulus near an outer edge of the island.Additional areas located throughout the film may also be bonded.

In an embodiment, the available active area is increased by theconstruction of compressively stressed membranes arranged such that thespacing between membranes is relatively small. Adjacent membranes may beformed from a continuous section of thin-film material. The dimensionsof the support structure are designed according to design rulespresented below so that the support structure is unlikely to buckle incompression.

A third structure in accordance with the invention combines the featuresof the above two structures to accommodate membranes which may be undertensile and compressive stress at various operating conditions, times,or locations. This structure is a small stressed membrane that has onlya slight overlap with the supporting structure, i.e., a support grid.The total dimensions of the stressed material, including freestandingand bonded areas, are designed to be sufficiently small so that themembrane is unlikely to fracture in tension according to the tensiledesign rules presented below. The allowable dimensions for the supportopening are based on the dimensions calculated using the compressivedesign rules presented below. Materials at the bond interface, i.e., thesupport grid, affect the characteristic crack spacing, and thereforealso affect the allowed dimensions of the membrane.

The membranes of the invention may be designed to survive repeatedthermal cycling. These membranes may be, for example,micro-electro-mechanical system (MEMS) based solid-oxide fuel cellmembranes.

In an aspect, the invention features a structure including a supportdefining a first opening, and a first tensilely stressed thin-filmmembrane disposed to occlude the first opening, the first stressedthin-film membrane contacting at least a first portion of the support.The first tensilely stressed thin-film membrane includes a membranematerial having a characteristic crack spacing greater than one-half ofa minimum dimension of the first stressed thin-film membrane and lessthan ten times the minimum dimension.

One or more of the following features may be included. The support maydefine a second opening adjacent to the first opening, the structurealso including a second tensilely stressed thin-film membrane disposedto occlude the second opening, the second stressed thin-film membranecontacting at least a second portion of the support. The secondtensilely stressed thin-film membrane may include the membrane materialand the characteristic crack spacing is greater than one-half of aminimum dimension of the second stressed thin-film membrane and lessthan ten times the minimum dimension of the second tensilely stressedthin-film membrane. A distance between the first and second openings maybe less than the minimum dimension of each opening.

The membrane may be disposed in an array and the array may include aplurality of stressed thin-film membranes and openings. A shape of theopening may be hexagonal, square, triangular, or circular. Across-sectional portion of the support may define a first shelf and anextension, and the stressed thin-film membrane may contact a portion ofthe first shelf. The cross-sectional portion of the support may define asecond shelf disposed in parallel to the first shelf, and the stressedthin-film membrane may contact a portion of the second shelf. Thecharacteristic crack spacing may be less than 1 mm. The stressedthin-film membrane may be disposed in an electrochemical system, e.g., asolid oxide fuel cell, or in a membrane-based hydrogen separationsystem.

The stressed thin-film membrane may include a material such as copper,nickel, palladium, platinum, rhenium, silicon carbide, aluminum nitride,an oxide, and/or combinations thereof. The oxide may be, e.g., an oxideof aluminum, cerium, chromium, cobalt, hafnium, iron, lanthanum,magnesium, manganese, samarium, scandium, silicon, strontium, titanium,ytterbium, yttrium, zirconium, praseodymium, and/or combinationsthereof.

In another aspect, the invention features a structure including asupport defining a first opening having a minimum opening dimension, anda first compressively stressed thin-film membrane disposed to occludethe first opening, the first stressed thin-film membrane contacting atleast a first portion of the support. The first compressively stressedthin-film membrane includes a membrane material, a critical aspect ratiofor buckling of the membrane material is greater than a ratio ofone-half of the minimum dimension of the first opening to a thickness ofthe stressed thin-film membrane, and the critical aspect ratio forbuckling is less than a ratio of ten times the minimum dimension of thefirst opening to the thickness of the stressed thin-film membrane.

One or more of the following features may be included. The support maydefine a second opening adjacent to the first opening, and a secondcompressively stressed thin-film membrane may be disposed to occlude thesecond opening, the second stressed thin-film membrane contacting atleast a second portion of the support. The second compressively stressedthin-film membrane may include the membrane material, a critical aspectratio for buckling of the membrane material may be greater than a ratioof one-half of a minimum dimension of the second opening to a thicknessof the stressed thin-film membrane, the critical aspect ratio forbuckling may be less than a ratio of ten times the minimum dimension ofthe first opening to the thickness of the stressed thin-film membrane,and the critical aspect ratio for buckling may be less than a ratio often times the minimum dimension of the second opening to the thicknessof the stressed thin-film membrane.

A distance between the first and second openings may be less than theminimum dimension of each opening.

The membrane may be disposed in an array, the array including aplurality of first stressed thin-film membranes and openings, and acritical aspect ratio for buckling of the array is less than the ratioof a minimum dimension of the array to the effective array thickness.The critical aspect ratio for buckling of the membrane material may beless than 40:1. The shape of the opening may be, e.g., hexagonal,square, triangular, or circular. A cross-sectional portion of thesupport may define a first shelf and an extension, and the stressedthin-film membrane may contact a portion of the first shelf. Thecross-sectional portion of the support may define a second shelfdisposed in parallel to the first shelf, and the stressed thin-filmmembrane may contact a portion of the second shelf.

The characteristic crack spacing may be less than 1 mm. The stressedthin-film membrane may be disposed in an electrochemical system, e.g., asolid oxide fuel cell or a membrane-based hydrogen separation system.

The stressed thin-film membrane may include a material such as copper,nickel, palladium, platinum, rhenium, silicon carbide, aluminum nitride,an oxide, and/or combinations thereof. The oxide may be, e.g., an oxideof aluminum, cerium, chromium, cobalt, hafnium, iron, lanthanum,magnesium, manganese, samarium, scandium, silicon, strontium, titanium,ytterbium, yttrium, zirconium, praseodymium, and combinations thereof.

In another aspect, the invention features a structure including asupport defining a first opening, and a first stressed thin-filmmembrane comprising a membrane material disposed to occlude the firstopening, the first stressed thin-film membrane contacting at least afirst portion of the support. At a first operating condition, the firststressed thin-film membrane is tensilely stressed and the membranematerial has a characteristic crack spacing greater than one-half of aminimum dimension of the first stressed thin-film membrane and less thanten times the minimum dimension. At a second operating condition, thefirst stressed thin-film membrane is compressively stressed and acritical aspect ratio for buckling of the membrane material is greaterthan a ratio of one-half of a minimum dimension of the first opening toa thickness of the stressed thin-film membrane, and the critical aspectratio for buckling is less than a ratio of ten times the minimumdimension of the first opening to the thickness of the stressedthin-film membrane.

In another aspect, the invention features a method of forming thestructures described above, including the steps of forming a supportdefining an opening, and forming a stressed thin-film membrane toocclude the opening.

One or more of the following features may be included. A substrate maybe provided, forming the support includes forming the support in atleast a region of a substrate, and the stressed thin-film membranecontacts both the support and the substrate. At least a portion of thesubstrate may be removed. An additional material may be deposited ontothe stressed thin-film membrane.

Forming the support may include forming a sacrificial layer over thesubstrate, defining a cavity in the substrate and the sacrificial layer,at least partially filling the cavity with a support material, andremoving at least a portion of the sacrificial layer to expose at leasta portion of a top surface of the support material.

The foregoing, and other features and advantages of the invention, aswell as the invention itself, will be more fully understood from thedescription and drawings that follow.

BRIEF DESCRIPTION OF DRAWINGS

FIGS. 1-11G are schematic cross-sectional and top views of thefabrication of an embodiment of the invention and schematic top views ofmasks used therein;

FIG. 12 is a schematic cross-sectional view of an embodiment of theinvention;

FIGS. 13 a-13 b are a schematic view of an embodiment of the inventionin use with an electrochemical system and a solid oxide fuel cell; and

FIG. 14 is a schematic view of an embodiment of the invention in usewith a hydrogen separation system.

The drawings are not necessarily to scale, emphasis instead generallybeing placed upon illustrating the principles of the invention. Theadvantages of the invention can be better understood by reference to thedescription taken in conjunction with the accompanying drawings.

DETAILED DESCRIPTION

A stressed thin-film structure may be formed in accordance with thefollowing process flow. Referring to FIG. 1, a substrate 10 may includea semiconductor material, such as double-sided polished silicon, and mayhave a diameter of, e.g., about 100 mm (not shown) and a thickness t₀of, e.g., about 50-500 μm. A sacrificial layer 20 is formed over thesubstrate 10. The sacrificial layer may include a dielectric material,such as silicon dioxide (SiO₂). In an embodiment, the sacrificial layer20 may be grown on both a front side 12 and a backside 14 of substrate10 by, e.g., steam oxidation. Steam oxidation may be performed in, forexample, in a furnace system available from Tystar Corporation, based inTorrance, Calif., in a steam ambient at 1050° C. The sacrificial layer20 may have a thickness t₁ sufficiently thick to enable the subsequentformation of ribs 120 (see, e.g., FIG. 8) having a height at least equalto a thickness of the subsequently formed stressed thin-film membrane190 (see, e.g., FIG. 11A). Thickness t₁ of the sacrificial layer may be,for example, approximately 2 μm.

Referring to FIGS. 2A-2D, a grid pattern 30 is defined in sacrificiallayer 20 and substrate 10 by, e.g., photolithography and etching. Aphotoresist layer 40 is spun on, exposed, and developed over thesacrificial layer 20. The photoresist layer has a thickness t₂ that issufficiently thick to withstand further processing, e.g., t₂ may beabout 2 μm. The grid pattern 30, as initially defined by patterning thephotoresist layer 40, may include an opening 45 having a width w₁ of,e.g., about 1.5 μm. The width w₁ of the opening 45 is selected such thatthe subsequently defined support grid structure 90 (see, e.g., FIG. 11A)will provide sufficient rigidity without drastically reducing the activearea of membranes 190 (see, e.g., FIG. 11A).

The grid pattern 30 is defined by using a grid mask 46 to pattern thephotoresist layer 40. The grid mask 46 includes a mask grid pattern 47defining a plurality of cells 48. The mask grid pattern may have lengthl₀ of, e.g., about 750 μm and a height h₀ of, e.g., about 750 μm. Eachcell may have a geometric shape, i.e., a hexagon with a distance d₀between parallel sides of, e.g., 10 to 40 μm, in accordance with thedesign rules presented below. A plurality of cells may form a honeycombpattern, as defined by the mask grid pattern. The grid mask may define adie having a height h₁ and a length l₁ of, e.g., about 10 mm each.

Referring also to FIG. 3, after the photoresist layer 40 is defined,portions 50 of the sacrificial layer 20 exposed by openings 45 in thephotoresist layer 40 are removed by, for example, dry etching. Dryetching may be performed by, for example, reactive ion etching (RIE) inan etching system such as the AMT 8100 system, manufactured by AppliedMaterials, Santa Clara, Calif., with an etching recipe appropriate forthe composition of the sacrificial layer, e.g., an oxide etch recipe.This etch may be an anisotropic etch that defines a plurality ofopenings 60 in the sacrificial layer 20 having a width w₂ that isapproximately equal to w₁, e.g., about 1.5 μm.

Referring to FIG. 4, after etching through sacrificial layer 20, anisotropic etch is performed, extending into substrate 10 to form aplurality of depressions 70. In an embodiment in which the substrate isformed of Si, this isotropic etch may be an sulfur hexafluoride (SF₆)etch performed for about 60-300 seconds by, e.g., a Multiplex system,manufactured by Surface Technology Systems, Wales, United Kingdom. Thisetch step defines depressions 70 in a substrate, where the depressionshave a depth d, of, e.g., about 3-4 μm and a width W₃ of about 8 μm.Depth d₁ and width w₃ are selected such that the support grid 90,subsequently formed in part in depressions 70 (see FIG. 11A), providesadequate support for subsequently formed stressed thin-film membranes190 (see FIG. 11A) without excessively reducing an active area of thesemembranes 190.

Referring to FIG. 5, the isotropic etch into substrate 10 is followed byan anisotropic etch to define a plurality of extensions 80 of thedepressions 70. Extensions 80 may be trenches defined in the substrate.The extensions may be formed by, e.g., an anisotropic etch in a systemsuch as the Multiplex system. In an embodiment in which the substrate 10contains silicon, an appropriate etch may be a recipe that uses an SF₆and octafluorocyclobutane (C₄F₈) chemistry. A method for anisotropicallyetching silicon is described in, for example, U.S. Pat. No. 5,501,893.Extensions 80 may each have a depth d₂ of, e.g., about 30-40 μm. Thedepth d₂ of extensions 80 is selected such that the materialsubsequently deposited in extensions 80 (see below) provides sufficientsupport, using the design rules for compressively stressed thin-filmmembranes presented below, to the subsequently formed stressed thin-filmmembrane, but sufficiently shallow such that gas flow to the membrane isnot hindered. For some embodiments with tensilely stressed thin-filmmembranes, the extension may not be needed. After these etch steps, thephotoresist layer 40 is stripped. The substrate 10 may then be cleanedby, e.g., an RCA clean followed by, e.g., a one-hour oxidation step at800° C. to eliminate any residual polymer. The grid pattern 30,initially defined by photoresist 40, is now defined by openings 60,depressions 70, and extensions 80.

Referring to FIG. 6, a support grid 90 is defined by deposition of asupport material 100 that may be, for example, a dielectric such assilicon-rich silicon nitride or TiO₂, over sacrificial layer 20 and intogrid pattern 30, including openings 60, depressions 70, and extensions80. The support material 100 may be deposited by, for example, chemicalvapor deposition (CVD), e.g., low-pressure CVD (LPCVD) orplasma-enhanced CVD (PECVD). The support material 100 may be under lowstress, e.g., <300 MPa, and may have a thickness t₃ of, e.g., about 2μm. The compressive stress of the support material 100 should besufficiently low to prevent buckling according to the compressive designrules presented below. The tensile stress is limited by the tensilestrength of the support material 100. The support grid 90 may be formedfrom a dielectric material to prevent electrical shorts in the finalstructure, e.g., in a fuel cell embodiment.

Referring to FIGS. 6 and 7, a top portion 105 of support material 100 isremoved by, e.g., a dry etch, such as a nitride etch in the AMT 8100system. Visual clearance endpoint detection may be used, with an overetch of approximately 5%.

Referring to FIGS. 7 and 8, removal of the top portion 105 of supportmaterial 100 exposes a portion 110 of sacrificial layer 20. Thesacrificial layer may be selectively removed by, for example, a wetetch. For example, in an embodiment in which the sacrificial layerincludes oxide, it may be removed by an oxide etch such as a bufferedoxide etch (BOE) that includes hydrofluoric acid, ammonia fluoride, andwater. The removal of the sacrificial layer may be followed by acleaning step such as an SC-1 (NH₄OH:H₂O₂) cleaning step, or an RCAclean followed by a one hour wet oxidation at 800° C. Removal of thesacrificial layer 20 exposes a portion of the support grid 90. Thisexposed portion includes rib 120 that has been formed by the depositionof support material 100 into opening 60 in sacrificial layer 20. Theribs 120 define generally the same pattern that had been originallydefined by mask 47, e.g., a plurality of hexagons. The ribs have aheight h₂ approximately equal to an initial thickness t₂ of thesacrificial layer, e.g., about 2 μm. The exposed portion of support grid90 forms a shelf 125.

Referring to FIG. 9, a stressed thin-film layer 130 is formed over theribs 120, exposed portions of front side 12 of substrate 10, and supportgrid 90, specifically over depressions 70 filled with support material100. The stressed thin-film layer may be, e.g., deposited, such as byelectron-beam evaporation. The stressed thin-film layer may include amembrane material that functions as an electrolyte material, forexample, YSZ, or may include a membrane material that functions as anelectrode, such as for example, a nickel/yttrium-stabilized zirconiacomposite. YSZ is a material particularly suitable for use as anelectrolyte in a solid-oxide fuel cell because it is a materialselectively permeable to oxygen ions at various partial pressures ofoxygen. In summary, some materials that may be used as a membranematerial to form the stressed thin-film layer include copper, nickel,palladium, platinum, rhenium, silicon carbide, aluminum nitride, anoxide such as an oxide of aluminum, cerium, chromium, cobalt, hafnium,iron, lanthanum, magnesium, manganese, samarium, scandium, silicon,strontium, titanium, ytterbium, yttrium, zirconium, praseodymium, and/orcombinations thereof. The stressed thin-film layer 130 may have athickness t₄ of, e.g., about 2 μm.

Depositing the stressed thin-film layer 130 over the ribs 120 may resultin a formation of intentional discontinuities 135 in the stressedthin-film layer 130 proximate the ribs 120. These intentionaldiscontinuities may be defined by controlling the step coverage of thestressed thin-film layer over the ribs. More specifically, nonconformalcoating by the stressed thin-film layer 130 leads to the formation ofdiscontinuities. The intentional discontinuities relieve stress in thestressed thin-film layer, thereby helping to prevent the formation ofunintentional cracks.

Referring to FIGS. 10A and 10B, a release photoresist pattern 140 isdefined over the backside 14 of substrate 10. As defined by a releasemask 150, the release photoresist pattern 140 may include a die 160having a length l₂ and height h₂ of, e.g., approximately 10 mm each. Acenter portion of the die may define a square 165 having sides with alength l₃ of, e.g., about 1.2 mm. Through use of mask 150, photoresistis patterned over substrate backside 14 such that the photoresist 140defines an open square 170 opposite support grid 90.

After the definition of release photoresist pattern, exposed portions ofsupport material 100 and sacrificial layer 20 are removed. Both layers100, 20 may be removed by, e.g., a dry etch in an AMT 8100 system fromApplied Materials.

Referring to FIGS. 11A and 11B, exposed portions of the substrate 10 areremoved by, for example, a wet etch. In an embodiment in which thesubstrate includes silicon, a suitable wet etch may be a potassiumhydroxide (KOH) etch. This composition selectively etches silicon alongcertain planes, resulting in a frame 200 in a shape of a square-based,flat-topped pyramid, i.e., an opening 201 created by this etch will bebroader at a bottom portion 202 at the backside 14 of substrate 10 andnarrower at a top portion 204 proximate the support grid 90. Forexample, if sides of the opening 170 defined by a release mask 150 havea length l₃ of, e.g., approximately 1.2 mm each, the opening proximatethe support grid 90 will have a length l₄ of, e.g., about 500 μm. Frame200 is formed from a same material as substrate 10, e.g., silicon, andincludes frame walls 205 having a thickness t₀ of, e.g., about 500 μm.

Referring to FIGS. 11A-11F, a result of the processing steps describedabove is a compound stressed thin-film membrane structure 175 having asupport grid 90 defining a plurality of openings 180. Each of theopenings 180 may have a hexagonal, square, triangular, or circularshape. A distance d₂₀ between first and second adjacent openings 180 a,180 b may be less than a minimum dimension d₃₀ of each opening.

The stressed thin-film layer 130 defines a plurality of stressedthin-film membranes 190, e.g., electrolyte layers, disposed to occludethe openings 180. The plurality of stressed thin-film membranes 190 mayinclude at least a first tensilely stressed thin-film membrane 190 a anda second tensilely stressed thin-film membrane 190 b, disposed toocclude a first opening 180 a and a second opening 180 b, respectively.Alternatively, the plurality of stressed thin-film membranes 190 mayinclude at least a first compressively stressed thin-film membrane 190 aand a second compressively stressed thin-film membrane 190 b, disposedto occlude a first opening 180 a and a second opening 180 b,respectively. The stressed thin-film membranes (also referred to as“tiles”) each contact at least a portion of the support grid 90. Thefirst tensilely or compressively stressed thin-film membrane may contactat least a first portion 195 a of the support and the second tensilelyor compressively stressed thin-film membrane may contact at least asecond portion 195 b of the support. The portion of the support grid 90contacted by the stressed thin-film membrane may be at least a portionof shelf 125.

The critical crack spacing and the critical aspect ratio for bucklingare the geometric values at which the probability of failure throughcracking or buckling is approximately equal to the probability of notcracking or buckling. In many practical applications, it is desirable tohave a very high probability of not failing. In these cases it may bebeneficial to add a factor of safety to the geometry. For example, thecritical crack spacing may be up to two times the minimum dimension ofthe stressed thin-film membrane. Alternatively, the critical crackspacing may be up to ten times the minimum dimension of the stressedthin-film membrane. In the case of compressive films, the criticalaspect ratio for buckling may be up to two times the ratio of theminimum opening dimension to the thickness of the stressed thin-filmmembrane. Alternatively, the critical aspect ratio for buckling may beup to ten times the ratio of the minimum opening dimension to thethickness of the stressed thin-film membrane.

These considerations may be taken into account in determining membranematerials and geometries. In some embodiments, each tensilely stressedthin-film membrane includes a membrane material having a characteristiccrack spacing that is greater than one-half of a minimum dimension ofthe membrane and less than ten times the minimum dimension. Thisrelationship may also be expressed as follows: the minimum dimension ofthe membrane is between twice and one-tenth the characteristic crackspacing. The membrane minimum dimension is determined by a minimumdistance between two sides of a shape defined by the membrane. Forexample, each stressed thin-film membrane may have the shape of ahexagon, having a minimum dimension equal to a distance d₀ betweenparallel sides of, e.g., about 20 μm, as defined by the cells 48 of gridmask 46 (see FIGS. 2B and 2C). Further discussion of characteristiccrack spacing is provided below in the discussion of the design rules.

The maximum dimension of the membrane may also be less than the criticalbuckling length. A compressively stressed thin-film membrane may includea membrane material having a critical aspect ratio for buckling that isgreater than a ratio of one-half of a minimum dimension of the firstopening to a thickness of the stressed thin-film membrane, and thecritical aspect ratio for buckling is less than a ratio of ten times theminimum dimension of the first opening to the thickness of the stressedthin-film membrane. This relationship may also be expressed as follows:a ratio of the minimum opening dimension to the thickness of thestressed-thin film membrane may be between twice and one-tenth of thecritical aspect ratio for buckling.

Further discussion of design rules for preventing buckling is providedbelow under the section heading “Overview of Design Rules forCompressive Stress.”

In some embodiments, the stressed thin-film membrane may be tensilelystressed at a first operating condition and the membrane material mayhave a characteristic crack spacing greater than one-half of a minimumdimension of the first stressed thin-film membrane. At a secondoperating condition, the first stressed thin-film membrane may becompressively stressed and a critical aspect ratio for buckling of themembrane material may be more than one-half of a minimum dimension ofthe first opening.

Referring to FIGS. 11B and 11G, compound stressed thin-film membranestructure 175 may be formed repeatedly across substrate 10, therebycreating a stressed thin-film membrane array consisting of two nestedand repeating lattice structures, i.e., a smaller lattice compoundstressed thin-film membrane structure 175 including the stressedthin-film membranes 190 disposed over support grid 90 and a largerlattice structure 220 including compound stressed thin-film membranes175 disposed over frame 200.

The support grid discussed above produces a support having a “t” shape.In an alternative embodiment, ribs may be omitted from the compoundstressed thin-film membrane structure by extending the removal of topportion 105 of support material 100 such that the rib is removed aswell. In this embodiment, therefore, the support grid may have a “T”shape.

It will be apparent to those skilled in the art that the support gridmay have a variety of cross-sectional shapes. The grid preferablyprovides a surface for attachment of the stressed thin-film membrane.Referring to FIG. 12, a cross-sectional portion of the support grid 90may define a first shelf 300 and an extension 310, and the tensilely orcompressively stressed thin-film membrane 190 may contact a portion 300a of the first shelf. The cross-sectional portion of the support mayalso define a second shelf 320 disposed in parallel to the first shelf,and the stressed thin-film membrane may contact a portion 320 a of thesecond shelf. In some embodiments, for example those with compressivefilms, the grid preferably has sufficient dimensions to provide rigidityto the membrane according to the compressive design rules providedbelow. It is found that a deeper grid provides more rigidity than awider grid. In a preferred embodiment for compressive stressed thin-filmmembranes, the ratio of depth to width of the grid is greater than 10.In some embodiments, for example those with tensilely stressed films,the grid may not provide any significant rigidity. For example, the gridmay be a flat thin film which spans from one stressed thin-film membraneto an adjacent stressed thin-film membrane. In other embodiments,particularly with compressively-stressed membranes, discontinuities inthe membranes may not be needed and a single membrane may occlude morethan one opening, e.g., two or more openings may be occluded by a singlemembrane.

In some embodiments of the invention, the grid comprises a dielectric.For example, in a fuel cell embodiment, the grid may be selected to benon-conductive to prevent shorting of the anode and cathode. In someembodiments of the invention, the grid comprises a diffusion barrier.For example, in a hydrogen purification embodiment, the grid may beselected from a material with a low diffusion coefficient for gasses.

Referring to FIGS. 13 a and 13 b, in use, the stressed thin-filmmembrane 190 may be disposed in an electrochemical system 400. Anelectrochemical system includes at least a first electrode 410, a secondelectrode 420 and an electrolyte 430 arranged so that passage of currentbetween electrodes causes a chemical reaction to occur. The electrolytemay be defined by the stressed thin-film membrane 190. The interactionof the chemical species with the electrodes also causes a voltage to begenerated between the electrodes. Electrochemical systems can be used,for example, to generate power such as in a fuel cell. In otherapplications, electrochemical systems can be used to sense the presenceor concentration of various chemical species. The present invention isparticularly useful for electrochemical systems in which the first andsecond electrodes 410, 420 are in communication with both theelectrolyte 430 and a first fluid 440 and second fluid 450 respectively,and it is desirable to maintain separation of the first and secondfluid.

The stressed thin-film membrane may be disposed in a solid oxide fuelcell. Referring to FIG. 13 b, the first fluid 440 may be a fuel and thesecond fluid 450 may be an oxidant. The first electrode 410 may be ananode, the second electrode 420 may be a cathode, and the electrolyte430 may be a solid oxide fuel cell electrolyte defined by stressedthin-film membrane 190.

Referring to FIG. 14, alternatively, the stressed thin-film membrane 190may be disposed in a membrane-based hydrogen separation system 500. Amembrane-based hydrogen separation system may include at least a firstfluid 510, the membrane 190, and a second fluid 520, with the membraneseparating the first and second fluids. The first fluid 510 may includehydrogen and at least a first diluent, with a first ratio of a hydrogenconcentration to a first diluent concentration. The membrane 190 mayinclude a material selected to be more permeable to hydrogen than to thefirst diluent. This selective permeability allows the second fluid 520to have a second ratio of hydrogen to the first diluent that is higherthan the first ratio. In some cases of membranes with extremely highselectivity, the second ratio may approach infinity. In someapplications, a second diluent is added to the second fluid to transportthe hydrogen away from the membrane.

Design Rules

In many embodiments, a compound stressed thin-film membrane structuremay be subjected to a range of stresses during operation. For example,if the stress is caused in part by thermal expansion mismatch betweenstressed thin-film membranes and a support grid or between a compoundstressed thin-film membrane array and frame, the stress will vary withtemperature. In another embodiment, the stress may vary over time.

A design for a compound stressed thin-film membrane structure isprovided, such that the structure is stable throughout the potentialstress range. Generally, it is sufficient to design the structure to berobust at the extremes of the stress range. In embodiments with thermalexpansion mismatch-based stress, the extremes generally occur at thehighest and lowest operating temperatures. One object of this inventionis to produce a membrane that is robust in both compression and tension.

A compound stressed thin-film membrane structure under excessive stressmay fail, e.g., warp (i.e., buckle) or crack. Failure may occur bycompression through cracks that form as a result of membrane buckling,for example, in the case of a YSZ membrane on silicon, at highertemperatures. Failure may also occur by formation of tensile cracks. Inan embodiment with a YSZ stressed thin-film membranes and a siliconframe, failure by warping in compression at elevated temperaturegenerally may initially be more likely than failure by cracking intension. In this embodiment, however, tension cracking at lowertemperatures may become more likely after many hours of device operationbecause of stress relaxation at elevated temperatures.

The compound stressed thin-film membrane structure has two features thatmay increase the robustness of a stressed thin-film membrane, such as aYSZ membrane, to tensile and compressive stress. Embodiments thatinclude at least one of ribs (alternatively called “rails” or “ridges”)and film discontinuities may act like stress relief joints, thereby bothreducing the probability that new cracks will form and preventing cracksthat may form from jumping from one stressed thin-film membrane to anadjacent stressed thin-film membrane. Any crack that forms, therefore,is isolated to an individual stressed thin-film membrane, and isprevented from causing failure of the compound stressed thin-filmmembrane structure. The support grid imparts stiffness to the compoundstressed thin-film membrane structure. This additional stiffness may, insome embodiments, prevent or reduce warping of stressed thin-filmmembranes and the compound stressed thin-film membrane structures undercompressive stress.

In some embodiments, particularly with compressively stressed thin-filmmembranes, film discontinuities may not be needed to reduce the risk ofcracks. A single membrane may, therefore, occlude more than one opening,e.g., two or more openings may be occluded by a single membrane.

Truly robust compound stressed thin-film membrane structures may beachieved by the application of a set of geometric design rules intendedto prevent the occurrence of warping and cracking. These design rules,applicable to many geometries, have been developed based on a synthesisof knowledge gained experimentally from the compound stressed thin-filmmembrane structure having ribs, a support grid, and a plurality ofstressed thin-film membranes.

These design rules are useful for producing more robust membranes, forexample membranes that are better able to tolerate thermal cycling, thanmembranes produced without taking into consideration the relationshipsdescribed herein.

In embodiments with membranes having thermal expansion rate differentfrom that of the frame, thermal cycling of the stressed thin-filmmembranes may subject the membranes to alternating states of compressionand tension. For example, in an embodiment in which a substrate is madeof Si and the stressed thin-film membranes are formed from YSZ, thecoefficients of thermal expansions (CTE) differ significantly: the CTEof YSZ is approximately 10 μm/m/° C., while that of Si and Si-richsilicon nitride are approximately 4 μm/m/° C. Because the YSZ thermalexpansion coefficient is nearly three times larger than that of Si andSi-rich silicon nitride, a hot YSZ membrane expands against theconstraint of its relatively fixed support grid and Si frame and is in astate of compression. Conversely, a membrane that has been cooled from arelaxed state at its operational temperature will be stretched by thesupport grid and Si support frame into a state of tension.

The stress difference between room temperature (30° C.) and operatingtemperature of 800° C. may be expressed as:

$\begin{matrix}\begin{matrix}{{Stress} = {E\left( {{\alpha_{X}\left( {T_{2} - T_{1}} \right)} - {\alpha_{Y}\left( {T_{2} - T_{1}} \right)}} \right)}} \\{= {E\left( {{\alpha_{YSZ}\left( {800 - 30} \right)} - {\alpha_{Si}\left( {800 - 30} \right)}} \right)}} \\{\approx {1200\mspace{20mu} {megapascals}\mspace{14mu} ({MPa})}}\end{matrix} & (0)\end{matrix}$

where E=Young's Modulus of the membrane,

-   -   α=coefficient of thermal expansion, and    -   T=temperature.

In an embodiment, the compound stressed thin-film membrane structure canwithstand a 2000 MPa change in the stress state of the stressedthin-film membrane without failure. In some embodiments, the compoundstressed thin-film membrane structure can withstand tensile stresses ashigh as 800 MPa. In still other embodiments, the compound membranestructure remains free of bow when it is in a compressive state as highas 1200 MPa.

As described above, the stressed thin-film membrane structure caninclude at least two nested, repeating lattice composite structures toprovide adequate stiffness or rigidity to the stressed thin-filmmembrane. The two nested structures may be built with different lengthscales, with different materials and material thicknesses, defined insuccessive fabrication steps.

The repeat distance or diameter of the smaller of the two lattices maybe only 10 to 40 μm, with a smallest repeating unit being a “cell.” Thecell may have a hexagonal shape and includes a free-standing YSZ thinfilm plate or “tile” (also referred to as “stressed thin-film membrane”)along with its mechanical frame or support grid. YSZ thin films aretypically 0.25-2 μm thick. The support grid is a mechanical supportstructure that holds the YSZ tiles. The cell wall, i.e., a portion ofthe support grid, may be formed from a dielectric such as silicon-richsilicon nitride and may have a width of 1-3 μm and a depth of 30-150 μm.The support grid may also include a shelf that may have a width of 1-5μm. The silicon-rich silicon nitride lattice may form a flat openhoneycomb structure, with a side, i.e., a top, of each cell sealed withthe thin film YSZ.

The compound stressed thin-film membrane structure may be a close-packed2-dimensional cell-array of the stressed-thin film membranes. It is thesmallest repeat unit of a larger structure, i.e., a compound stressedthin-film membrane array. The compound stressed thin-film membrane arraymay have a diameter of, for example, 5-100 mm, with each of the compoundstressed thin film membranes having a diameter between 200 μm to 2 mm.Each member of the compound stressed thin-film membrane array includesits own compound stressed thin-film membrane plus an additionalintervening mechanical structure—the “membrane wall” frame. In anembodiment, the frame may be composed of silicon with a thickness of 50to 500 μm. The thicker frame walls are defined in a processing step,e.g., an etch step, separately from the formation of the cells and cellwalls, i.e., the membranes and the support grid.

Two methodologies are presented for determining advantageous dimensionalrelationships between the thickness of the stressed thin-film membranetile material, e.g., YSZ, the thickness and depth of the cell wall,e.g., silicon-rich silicon nitride, the diameter of a cell, thethickness and depth of the membrane wall, and the diameter of thecompound membrane. The first set of relationships applies to embodimentsthat experience significant tensile stresses. The second set ofrelationships applies to embodiments that experience significantcompressive stresses. Application of the relationships applicable tocases of significant compressive, tensile, or both compressive andtensile stresses enables the formation of stressed thin-film membranesand compound stressed thin-film membrane arrays substantially free ofwarp and cracking.

Overview of Design Rules for Tensile Stress

A maximum dimension a tensilely stressed thin-film membrane may havewithout cracking is determined by the characteristic crack spacing ofthe film. A tensilely stressed thin film is defined here as a film thatwill spontaneously develop cracks when the lateral dimensions of thefilm are sufficiently large. The occurrence of cracks is dependent upon,for example, the internal stress state of the film, the thickness of thefilm, the roughness of each surface of the film, the adhesion of thefilm to the substrate, the composition, and the frequency and characterof film defects.

When a crack forms in a tensilely stressed thin-film, local stress inthe film is partially relieved, thereby reducing the probability of anadditional crack forming near the first crack. The combined effect ofthe global tendency towards cracking and the local stress reliefresulting from a crack generally causes the distance betweensubstantially parallel cracks to cluster around a characteristic crackspacing, with some scatter around that characteristic crack spacing dueto random processes. This crack formation has the appearance of a driedand cracked lakebed with relatively uniform sizes of un-cracked areas.

In some embodiments, the characteristic crack spacing may beanisotropic, with certain directions cracking at a higher frequency thanother directions. One technique for quantifying the directionalcharacteristic crack spacing is as follows:

-   (A) Obtain a micrograph of the film by, e.g., a dark-field optical    microscope.-   (B) Select a sampling region sufficiently large to be representative    of the film. The area may extend at least 10 crack spacings in each    direction, and preferably at least 100 crack spacings.-   (C) If there is a crack in the sampling region that does not    terminate at another crack, i.e., has a free end, with a pen, extend    the crack until it intersects another crack.-   (D) Find the center of gravity of each island of film. An island is    defined as an area of film that is surrounded by cracks (and    extended cracks from C), and that does not contain any cracks. To    find the center of gravity, one technique is to bisect each island    with a straight line such that the areas on each side of the line    are equal. Repeat the bisection with a perpendicular line. The    intersection of the two bisection lines is the center of gravity of    the island.-   (E) Label each island with a uniform coordinate system so that the    absolute orientation of the islands can be maintained. Section the    micrograph along all of the cracks so that the islands can be moved    independently. Superimpose all of the islands of film aligning the    center of gravity of each film, while maintaining the absolute    orientation of each island.-   (F) In each direction, the characteristic crack spacing may be found    by averaging the diameter of the islands along the selected axis.    The standard deviation may also be found along the axis.

Preferably the diameter of the island, e.g., a stressed thin-filmmembrane, is less than twice the crack spacing in all directions. Morepreferably, the diameter of the cell is less than one-half of the crackspacing in all directions.

The dimensions of the tensilely stressed thin-film membrane are definedby the edges of the membrane. The edges may be defined by anyintentional non-uniformity in the membrane that prevents orsubstantially reduces the transfer of stress across the non-uniformityregion. Most preferably, the edges are defined by a discontinuity in thestressed thin-film membrane material. Alternatively, the edge may be astress-relief joint or a crack-inducing shape.

The invention preferably separates each island in such a way as toreduce the frequency of crack formation in each island and to reduce thefrequency of crack propagation between adjacent islands.

One structure in accordance with the invention includes stressedthin-film membrane islands that are discontinuous. The distance betweenadjacent islands is selected such that the islands do not come incontact during at least some portion of time while operating. The spacebetween adjacent islands may be formed in some embodiments byphotopatterning and etching. In another embodiment, the space may befilled with another material, for example low-stress silicon nitride. Inanother embodiment, the space may be formed by including a rail over thesupport grid and eliminating the step coverage during deposition of thestressed thin-film material, resulting in a discontinuous film over therail.

Another structure in accordance with the invention includes astress-relief joint between adjacent islands. In one embodiment, jointmay be in the form of a “U” such that the arms of the “U” can movetogether or apart to relieve stress. In another embodiment, the jointmay be formed from a material other than a stressed thin-film material.

A third structure in accordance with the invention includes a featuredesigned to cause the stressed-thin film to crack in a controlledlocation. In one embodiment, the film may be deposited continuously overa step feature. Cracks will preferentially form parallel to the step andwithin a distance from the step approximately equal to the filmthickness.

Preferably the compound stressed thin-film membrane includes a structurethat separates each island to reduce the frequency of crack formation ineach island and to reduce the frequency of crack propagation betweenadjacent islands.

Overview of Design Rules for Compressive Stress

A maximum dimension that a stressed thin-film membrane may have withoutbuckling may be determined by applying an Euler's formula for thebuckling of thin plates (see equation 1 below). This equation yields arelationship between the thickness of the stressed thin-film membrane(t), a length of the stressed thin-film membrane (L in the instance of asquare cell), and maximum tolerable compressive stress (σ). This rule isintended to ensure that the stressed thin-film membranes remain free ofbow, i.e., will not buckle.

In an embodiment, a stressed thin-film membrane has a square shape,supported only by its edges. Here a critical aspect ratio for buckling,i.e., a minimum length to thickness ratio that may be tolerated beforebuckling occurs, may be calculated by applying an Euler formula, e.g.,Equation 1:

$\begin{matrix}{\frac{L}{t} = \sqrt{\frac{\pi^{2}E}{3\left( {1 - v^{2}} \right)\sigma}}} & (1)\end{matrix}$

where L=Length of the square,

t=Thickness of the film,

v=Poisson ratio of the film,

σ=Compressive stress in the film, and

E=Young's modulus of the film:

This formula is relatively conservative in that tiles and membranes arefrequently at least partially clamped at their edges. The criticalaspect ratio will be larger for clamped edges, so this ratio of L/t maybe considered to be a lower bound.

This Euler's formula assumes a square shape. The critical aspect ratiofor hexagons and other shapes varies somewhat, but is expected to bewithin 30% of this value.

A simplification of the rule may be that, to avoid buckling, a ratio ofcell diameter to membrane film thickness should generally be not greaterthan 20. This value is based on the assumptions that most stressed filmshave internal stress of <1 GPa, most Young's moduli of brittle films are˜150 GPa, and most Poisson ratios are ˜0.25. In some cases the ratio maybe as large as 40.

In the case of a compound stressed thin-film membrane, a maximumdimension may also be determined by an Euler formula for the buckling ofa thin composite plate. One may use Equation 1 below, with the effectivevariables for the structure under consideration. The equation provides arelationship between the flexural rigidity of the membrane (thestiffness or effective thickness of the cell materials, i.e., the wallsand tiles), the membrane diameter, and the maximum tolerable compressivestress. This rule is intended to ensure that the compound stressedthin-film membrane will not buckle. The effective membrane thickness andmodulus are determined, primarily, by the dimensions of the silicon-richsilicon nitride cell walls. The three factors that determine the ratioof the component materials parameters and the effective membranethickness and modulus are the support grid width, the support griddepth, and cell diameter.

The cell support grids may be designed with a particular height ortopography (e.g., with ribs) to break the planar continuity of thestressed thin-film membrane. The purpose of the rib is to create adiscontinuity in the stressed thin-film layer that allows for somestress relaxation of the stressed thin-film material and acts as atermination point for any unintended cracks in the stressed thin-filmmaterial. This height is selected to be similar to the film thickness,e.g., 2 μm.

In the case of a compressively stressed thin-film membrane array, thedimensions of the frame, e.g., silicon walls, may be set by yet anotherapplication of an Euler formula. Again, one applies Equation 1, with anallowance for the shape of each compound membrane. This equationprovides a relationship between the flexural rigidity of the completecompound stressed thin-film membrane array (the effective thickness ofall membrane, grid, and frame materials), the diameter of the completecompound stressed thin-film array, and the maximum tolerable compressivestress. This rule is intended to ensure that the compound stressedthin-film membrane array will not bow, i.e., will not buckle. To ensurerelatively high yields, this aspect ratio may be less than 2 times theminimum dimension of the first opening to the membrane thickness. Theeffective array thickness and modulus are primarily determined by thedimensions of the membrane walls. The three factors that determine theratio of the component materials parameters and the effective arraythickness and modulus are the membrane wall width, the membrane wallthickness, and membrane diameter.

To apply Equation 1 to a YSZ stressed thin-film membrane, one may usethe following approximate values:

E=160 GPa

v=0.23

σ=1 GPa

Using the above Equation 1, the critical aspect ratio, i.e., the maximumlength to thickness ratio one may use before buckling occurs is:

L/t=23.6

Therefore, in an embodiment in which the thickness of the stressedthin-film membrane is 2 μm, one may form tiles with a maximum width 47μm without buckling.

For the compound stressed thin-film membrane (many tiles), thecalculation of maximum dimensions possible prior to occurrence offailure is more difficult because the membrane is no longer a solidpiece of material. The actual materials parameters in Equation 1 must besubstituted with effective parameters that include the effects ofgeometry, stress and materials parameters of the components of thecompound stressed thin-film membrane. For example, the effective Young'smodulus for a hexagonal honeycomb is:

$\begin{matrix}{E^{*} = {{E \cdot 2.3}\left( \frac{w}{a} \right)^{3}}} & (2)\end{matrix}$

where w=Width of the walls

a=Length of each wall

See Cellular Solids by L. J. Gibson and M. F. Ashby (Second Edition,1997).

The effective compressive stress will also be reduced by the thicknessof the membrane:

$\begin{matrix}{\sigma^{*} = {\sigma \frac{b}{t}}} & (3)\end{matrix}$

where b=Thickness of the YSZ

t=Thickness of the honeycomb

This formula is relatively conservative in that it does not include anycontribution to membrane strength from the YSZ tiles or the “shelf.”However, it also does not take into account that the stress is primarilyapplied to the top of the membrane, which will tend to encouragebuckling.

A characteristic hexagonally shaped membrane has the followingproperties:

E=160 GPa

v=0.25=

σ=1 GPa

t=40 μmw=1.5 μmb=2 μma=20 μm/sqrt(12)=5.8 μm (This factor converts the diameter of a hexagoninto the length of a side)

σ*=50 MPa

E*=6.45 GPa

By substituting the starred quantities into Equation 1, one obtains

L/t=21.3

L=0.85 mm

Note: Other honeycomb geometries, for example, triangular cells, mayhave very different effective Young's Moduli. As an example, theequation for the effective Young's modulus for triangular cells is:

$\begin{matrix}{E^{*} = {{E \cdot 1.15}\left( \frac{w}{a} \right)}} & (4)\end{matrix}$

See Cellular Solids. For a YSZ membrane, E*=52 GPa, and L/t=68, yieldinga maximum dimension of L=3.4 mm in an embodiment with a triangular cell.

Despite the possibility provided by a triangular shape of having alarger maximum dimension before the occurrence of buckling, incomparison to a hexagonal shape, hexagonally shaped stressed thin-filmmembranes may be preferable. The latter provide a higher ratio of activearea to support grid area than is provided by honeycomb structures withtriangular cells.

It is noted that fabrication of stressed thin-film membranes inaccordance with the methods described above has resulted in increases ofyield of about 80% for membranes subjected to thermal cycling up to 800°C.

Throughout the description, where compositions are described as having,including, or comprising specific components, or where processes aredescribed as having, including, or comprising specific process steps, itis contemplated that compositions of the present invention also consistessentially or, or consist of, the recited components, and that theprocesses of the present invention also consist essentially of, orconsist of, the recited processing steps.

It should be understood that the order of steps or order for performingcertain actions is immaterial so long as the invention remains operable.Moreover, two or more steps or actions may be conducted simultaneously.

The invention may be embodied in other specific forms without departingfrom the sprit or essential characteristics thereof. The foregoingembodiments are therefore to be considered in all respects illustrativerather than limiting on the invention described herein.

Each of the patent documents and scientific publications disclosedhereinabove is incorporated by reference herein for all purposes.

1-32. (canceled)
 33. A fuel cell assembly comprising: a membraneassembly having a support defining an opening; and a stressed thin-filmmembrane coupled to at least one portion of the support, the stressedthin-film membrane occluding the opening, the stressed thin-filmmembrane having a thickness and having a shape, the shape having adimension of less than about 80 microns.
 34. The fuel cell assembly ofclaim 33 further comprising: first and second electrodes, the membraneassembly at least partially disposed between the first and secondelectrodes.
 35. The fuel cell assembly of claim 33, wherein the shapehas a dimension of less than about 40 microns.
 36. The fuel cellassembly of claim 33, wherein the support defines a plurality ofopenings, and the membrane assembly includes a plurality of stressedthin-film membranes, each stressed thin-film membrane coupled to atleast one portion of the support and occluding a corresponding opening,and each thin-film membrane having a shape, each shape having adimension of less than about 80 microns.
 37. The fuel cell assembly ofclaim 36, wherein a distance between two adjacent openings is less thanabout 80 microns.
 38. The fuel cell assembly of claim 36 furthercomprising: a frame, wherein the membrane assembly is coupled to atleast one portion of the frame.
 39. The fuel cell assembly of claim 38further comprising: a plurality of membrane assemblies, wherein eachmembrane assembly is coupled to at least one portion of the frame. 40.The fuel cell assembly of claim 33, wherein the support includes a firstshelf, and the stressed thin-film membrane is coupled to at least oneportion of the first shelf.
 41. The fuel cell assembly of claim 40,wherein the support includes a second shelf disposed approximatelyparallel to the first shelf, and the stressed thin-film membrane iscoupled to at least one portion of the second shelf.
 42. The fuel cellassembly of claim 33, wherein the support includes at least one rib, thethin-film membrane adjacent to the at least one rib.
 43. The fuel cellassembly of claim 33, wherein the shape of the thin-film membrane isselected from the group consisting of hexagonal, square, triangular, andcircular.
 44. The fuel cell assembly of claim 33, wherein the openinghas a shape, and the shape of the opening is selected from the groupconsisting of hexagonal, square, triangular, and circular.
 45. A methodof forming a fuel cell assembly, the method comprising: forming asupport defining an opening; and forming a stressed thin-film membraneon at least one portion of the support, the stressed thin-film membraneoccluding the opening and having a thickness and a shape, the shapehaving a dimension of less than about 80 microns.
 46. The method ofclaim 45 further comprising: providing first and second electrodes, thesupport and the stressed thin-film membrane at least partially disposedbetween the first and second electrodes.
 47. The method of claim 45,wherein the shape has a dimension of less than about 40 microns.
 48. Themethod of claim 45, wherein the support defines a plurality of openings,and the method includes forming a plurality of stressed thin-filmmembranes, each stressed thin-film membrane formed on at least oneportion of the support and occluding a corresponding opening, and eachthin-film membrane having a shape, each shape having a dimension of lessthan about 80 microns.
 49. The method of claim 48, wherein a distancebetween two adjacent openings is less than about 80 microns.
 50. Themethod of claim 48, further comprising: forming a frame, wherein atleast one portion of the support or the stressed thin-film membrane isformed on at least one portion of the frame.
 51. The method of claim 45,wherein the support includes a first shelf, and the stressed thin-filmmembrane is formed on at least one portion of the first shelf.
 52. Themethod of claim 51, wherein the support includes a second shelf disposedapproximately parallel to the first shelf, and the stressed thin-filmmembrane is formed on at least one portion of the second shelf.
 53. Themethod of claim 45, wherein the support includes at least one rib, andthe thin-film membrane is formed adjacent to the at least one rib. 54.The method of claim 45, wherein the shape of the thin-film membrane isselected from the group consisting of hexagonal, square, triangular, andcircular.